In this paper, we deal with some shadowing and chain recurrence properties for continuous map from a compact metric space to itself. We investigate the relation between a dynamical system  and its set-valued and fuzzified counterparts (i.e.,  and  respectively) in the related dynamical properties such as chain recurrence, chain transitivity, chain mixing, POTP and APOTP.

Zadeh’s extension, chain recurrence, chain transitivity, pseudo orbit tracing property, fuzzified dynamical system.